Geodesic
Normed spaces which satisfy Apollonius' theorem
Matematičeskie zametki, Tome 20 (1976) no. 2, pp. 247-252 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is proved that a normed space is a Hilbert space if it possesses the property: The geometric locus of the points, for which the ratio of the distances to two given points is constant, is a sphere. 
@article{MZM_1976_20_2_a9,
     author = {I. A. Danelich},
     title = {Normed spaces which satisfy {Apollonius'} theorem},
     journal = {Matemati\v{c}eskie zametki},
     pages = {247--252},
     year = {1976},
     volume = {20},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1976_20_2_a9/}
}
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I. A. Danelich. Normed spaces which satisfy Apollonius' theorem. Matematičeskie zametki, Tome 20 (1976) no. 2, pp. 247-252. http://geodesic.mathdoc.fr/item/MZM_1976_20_2_a9/